Chapter 1 Asset Returns The primary goal of investing in a financial market is to make profits without taking excessive risks. Most common investments involve purchasing financial assets such as stocks, bonds or bank deposits, and holding them for certain periods. Posi- tive revenue is generated if the price of a holding asset at the end of holding period is higher than that at the time of purchase (for the time being we ignore transaction charges). Obviously the size of the revenue depends on three factors: (i) the initial capital (i.e. the number of assets purchased), (ii) the length of holding period, and (iii) the changes of the asset price over the holding period. A successful investment pursues the maximum revenue with a given initial capital, which may be measured explicitly in terms of the so-called return . A return is a percentage defined as the change of price expressed as a fraction of the initial price. It turns out that asset returns exhibit more attractive statistical properties than asset prices themselves. Therefore it also makes more statistical sense to analyze return data rather than price series. 1.1 Returns Let Pt denote the price of an asset at time t. First we introduce various definitions for the returns for the asset. 1.1.1 One-period simple returns and gross returns Holding an asset from time t − 1tot, the value of the asset changes from Pt−1 to Pt. Assuming that no dividends paid are over the period. Then the one-period simple return is defined as Rt =(Pt − Pt−1)/Pt−1. (1.1) It is the profit rate of holding the asset from time t − 1tot.Oftenwewrite Rt = 100Rt%, as 100Rt is the percentage of the gain with respect to the initial capital Pt−1. This is particularly useful when the time unit is small (such as a day or an hour); in such cases Rt typically takes very small values. The returns for less 2 Chapter 1 Asset Returns risky assets such as bonds can be even smaller in a short period and are often quoted in basis points,whichis10, 000Rt. The one period gross return is defined as Pt/Pt−1 = Rt + 1. It is the ratio of the new market value at the end of the holding period over the initial market value. 1.1.2 Multiperiod returns The holding period for an investment may be more than one time unit. For any integer k 1, the returns for over k periods may be defined in a similar manner. For example, the k-period simple return from time t − k to t is Rt(k)=(Pt − Pt−k)/Pt−k, and the k-period gross return is Pt/Pt−k = Rt(k) + 1. It is easy to see that the multiperiod returns may be expressed in terms of one-period returns as follows: Pt Pt Pt−1 Pt−k+1 = ··· , (1.2) Pt−k Pt−1 Pt−2 Pt−k Pt Rt(k)= − 1=(Rt +1)(Rt−1 +1)···(Rt−k+1 +1)− 1. (1.3) Pt−k If all one-period returns Rt, ··· ,Rt−k+1 are small, (1.3) implies an approximation Rt(k) ≈ Rt + Rt−1 + ···+ Rt−k+1. (1.4) This is a useful approximation when the time unit is small (such as a day, an hour or a minute). 1.1.3 Log returns and continuously compounding In addition to the simple return Rt, the commonly used one period log return is defined as rt =logPt − log Pt−1 = log(Pt/Pt−1) = log(1 + Rt). (1.5) Note that a log return is the logarithm (with the natural base) of a gross return and log Pt is called the log price. One immediate convenience in using log returns is that the additivity in multiperiod log returns, i.e. the k period log return rt(k) ≡ log(Pt/Pt−k) is the sum of the k one-period log returns: rt(k)=rt + rt−1 + ···+ rt−k+1. (1.6) An investment at time t − k with initial capital A yields at time t the capital kr¯ A exp{rt(k)} = A exp(rt + rt−1 + ···+ rt−k+1)=Ae , 1.1 Returns 3 wherer ¯ =(rt + rt−1 + ···+ rt−k+1)/k is the average one-period log returns. In this book returns refer to log returns unless specified otherwise. Note that the identity (1.6) is in contrast with the approximation (1.4) which is only valid when the time unit is small. Indeed when the values are small, the two returns are approximately the same: rt = log(1 + Rt) ≈ Rt. However, rt <Rt. Figure 1.1 plots the log returns against the simple returns for the Apple Inc share prices in the period of January 1985 – February 2011. The returns are calculated based on the daily close prices for the three holding periods: a day, a week and a month. The figure shows that the two definitions result almost the same daily returns, especially for those with the values between −0.2 and 0.2. However when the holding period increases to a week or a month, the discrepancy between the two definitions is more apparent with a simple return always greater than the corresponding log return. Figure 1.1 Plots of log returns against simple returns of the Apple Inc share prices in January 1985 – February 2011. The blue straight lines mark the positions where the two returns are identical. The log return rt is also called continuously compounded return due to its close link with the concept of compound rates or interest rates . For a bank deposit account, the quoted interest rate often refers to as ‘simple interest ’. For example, an interest rate of 5% payable every six months will be quoted as a simple interest of 10% per annum in the market. However if an account with the initial capital $1 is held for 12 months and interest rate remains unchanged, it follows from (1.2) that the gross return for the two periods is 1 × (1 + 0.05)2 =1.1025, i.e. the annual simple return is 1.1025 − 1=10.25%, which is called the compound 4 Chapter 1 Asset Returns return and is greater than the quoted annual rate of 10%. This is due to the earning from ‘interest-on-interest’ in the second six-month period. Now suppose that the quoted simple interest rate per annum is r and is un- changed, and the earnings are paid more frequently, say, m times per annum (at the rate r/m each time of course). For example, the account holder is paid every quarter when m =4,everymonthwhenm = 12, and every day when m = 365. Suppose m continues to increase, and the earnings are paid continuously eventually. Then the gross return at the end of one year is lim (1 + r/m)m = er. m→∞ More generally, if the initial capital is C, invested in a bond that compounds con- tinuously the interest at annual rate r, then the value of the investment at time t is C exp(rt). Hence the log return per annum is r, which is the logarithm of the gross return. This indicates that the simple annual interest rate r quoted in the market is in fact the annual log return if the interest is compounded continuously. Note that if the interest is only paid once at the end of the year, the simple return will be r,andthe log return will be log(1 + r) which is always smaller than r. In summary, a simple annual interest rate quoted in the market has two interpre- tations: it is the simple annual return if the interest is only paid once at the end of the year, and it is the annual log return if the interest is compounded continuously. 1.1.4 Adjustment for dividends Many assets, for example some blue-chip stocks, pay dividends to their share- holders from time to time. A dividend is typically allocated as a fixed amount of cash per share. Therefore adjustments must then be made in computing returns to account for the contribution towards the earnings from dividends. Let Dt denote the dividend payment between time t − 1andt. Then the returns are now defined as follows: Rt =(Pt + Dt)/Pt−1 − 1,rt = log(Pt + Dt) − log Pt−1, Rt(k)= Pt + Dt + ···+ Dt−k+1 Pt−k − 1, k−1 Pt−j + Dt−j rt(k)=rt + ···+ rt−k+1 = log . j=0 Pt−j−1 The above definitions are based on the assumption that all dividends are cashed out and are not re-invested in the asset. 1.1 Returns 5 1.1.5 Bond yields and prices Bonds are quoted in annualized yields. A so-called zero-coupon bond is a bond bought at a price lower than its face value (also called par value or principal), with the face value repaid at the time of maturity. It does not make periodic interest payments (i.e. coupons), hence the term ‘zero-coupon’. Now we consider a zero- coupon bond with the face value $1. If the current yield is rt and the remaining duration is D units of time, with continuous compounding, its current price Bt should satisfy the condition Bt exp(Drt)=$1, i.e. the price is Bt =exp(−Drt) dollars. Thus, the annualized log-return of the bond is log(Bt+1/Bt)=D(rt − rt+1). (1.7) Here, we ignore the fact that Bt+1 has one unit of time shorter maturity than Bt.